Hypothesis Testing with Two Samples: Dependent Samples (Paired Data)

Chapter 8: Hypothesis Testing with Two Samples

Section 8.3: Testing the Difference Between Means (Dependent Samples)

This section focuses on hypothesis testing for the difference between means when the samples are dependent, also known as paired samples. This is commonly used when the same subjects are measured twice (e.g., before and after a treatment), or when subjects are matched in pairs.

Key Concepts

• Dependent (Paired) Samples: Two samples are dependent if each value in one sample is paired with a corresponding value in the other sample. Examples include before-and-after measurements on the same subjects or matched pairs in experiments.

• Paired Differences: For each pair, calculate the difference between the two measurements. The analysis is then performed on these differences.

Conditions for the t-Test for Dependent Samples

• The samples must be randomly selected.

• The samples must be dependent (paired).

• Both populations must be normally distributed, or the number of pairs (n) must be at least 30 (Central Limit Theorem applies).

Symbols Used

• n: Number of pairs

• \( \overline{d} \): Mean of the differences between paired data entries

• \( s_d \): Standard deviation of the differences

• \( \mu_d \): Population mean of the differences

Hypotheses

• Null Hypothesis (\( H_0 \)): \( \mu_d = 0 \) (no difference in means)

• Alternative Hypothesis (\( H_a \)): \( \mu_d \neq 0 \), \( \mu_d > 0 \), or \( \mu_d < 0 \) depending on the claim

Test Statistic

The test statistic for the paired t-test is calculated as:

$t = \frac{\overline{d} - \mu_d}{s_d / \sqrt{n}}$

• Where \( \overline{d} \) is the sample mean of the differences, \( s_d \) is the sample standard deviation of the differences, and n is the number of pairs.

• The test statistic follows a t-distribution with \( n - 1 \) degrees of freedom.

Decision Rule

• Find the critical value(s) from the t-distribution table based on the significance level (\( \alpha \)) and degrees of freedom (\( n-1 \)).

• Compare the calculated t-value to the critical value(s):

  • If the test statistic falls in the rejection region, reject the null hypothesis.

  • If not, fail to reject the null hypothesis.

Example 1: Golf Scores Before and After Lessons

Scenario: A company claims that golf lessons decrease average golf scores. Eight golfers' scores are measured before and after lessons. Is there enough evidence at the 10% significance level to support the claim?

• Step 1: State hypotheses:

  • \( H_0: \mu_d = 0 \) (no improvement)

  • \( H_a: \mu_d > 0 \) (scores decrease after lessons)

• Step 2: Check conditions: Samples are random, paired, and normally distributed.

• Step 3: Calculate the differences (before - after) for each golfer, then compute \( \overline{d} \) and \( s_d \).

• Step 4: Compute the test statistic using the formula above.

• Step 5: Determine the critical value for a right-tailed test at \( \alpha = 0.10 \) and compare.

• Step 6: Conclusion: If t is not in the rejection region, fail to reject \( H_0 \). In this example, there is not enough evidence to support the company's claim at the 10% level.

Example 2: Legislator’s Performance Ratings

Scenario: A legislator wants to know if their performance rating has changed from last year to this year, based on ratings from 16 voters. Is there enough evidence at the 5% significance level to conclude a change?

• Step 1: State hypotheses:

  • \( H_0: \mu_d = 0 \) (no change)

  • \( H_a: \mu_d \neq 0 \) (there is a change)

• Step 2: Check conditions: Samples are random, paired, and normally distributed.

• Step 3: Calculate the differences (last year - this year) for each voter, then compute \( \overline{d} \) and \( s_d \).

• Step 4: Compute the test statistic using the formula above.

• Step 5: Determine the critical values for a two-tailed test at \( \alpha = 0.05 \) and compare.

• Step 6: Conclusion: If t is in the rejection region, reject \( H_0 \). In this example, there is enough evidence to conclude the rating has changed.

Summary Table: Steps for the Paired t-Test

Step	Description
1	State the null and alternative hypotheses
2	Check conditions (random, paired, normality or n ≥ 30)
3	Calculate differences, mean (\( \overline{d} \)), and standard deviation (\( s_d \))
4	Compute the test statistic: $t = \frac{\overline{d} - \mu_d}{s_d / \sqrt{n}}$
5	Find critical value(s) and rejection region(s)
6	Make a decision and state the conclusion

Additional info: The paired t-test is widely used in medical, psychological, and educational research to assess the effect of interventions or treatments when measurements are taken on the same subjects before and after the intervention.